MinT - Best Known (30−19, 30, s)-Nets in Base 81

Best Known (30−19, 30, s)-Nets in Base 81

(30−19, 30, 200)-Net over F₈₁ — Constructive and digital

Digital (11, 30, 200)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
      - a function field by SÃ©mirat [i]
2. digital (1, 20, 100)-net over F₈₁, using
  - net from sequence [i] based on digital (1, 99)-sequence over F₈₁ (see above)

(30−19, 30, 244)-Net over F₈₁ — Digital

Digital (11, 30, 244)-net over F₈₁, using

t-expansion [i] based on digital (9, 30, 244)-net over F₈₁, using
- net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(30−19, 30, 73147)-Net in Base 81 — Upper bound on s

There is no (11, 30, 73148)-net in base 81, because

1 times m-reduction [i] would yield (11, 29, 73148)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 22 186494 976048 479185 213842 275743 670288 230367 462138 656961 > 81²⁹ [i]